Pome

Corollary 15.27.605B92E5A4684

Corollary 15.27.6.05B92E5A4684 Assume $v_-(h)=0$, and let $\mu =f\, \mathrm{d}V$ be the measure of theorem 15.27.3. Then $f\notin L^ p(\mathrm{d}V)$ for every $p>1$. Moreover, for every probability density $g\in L^1(\mathrm{d}V)$ and every $0<t<1$, the probability density

\[ g_ t=(1-t)g+tf \]

admits no positive scaled subsolution, although $\| g_ t-g\| _{L^1}\leq 2t$. In particular, the probability densities admitting no positive scaled subsolution are dense among all probability densities in $L^1(\mathrm{d}V)$.

Proof. If $f\in L^ p$ for some $p>1$, theorem 14.19.2 would give a continuous solution of $\mathrm{MA}_ h(u)=cf\, \mathrm{d}V$ with $c>0$, contradicting the theorem. Since $g_ t\, \mathrm{d}V\geq t\mu $, a positive scaled subsolution for $g_ t\, \mathrm{d}V$ would also give one for $\mu $. Finally, $\| g_ t-g\| _{L^1}=t\| f-g\| _{L^1}\leq 2t$. $\square$

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